diff options
Diffstat (limited to 'frsb_kac-rice.tex')
| -rw-r--r-- | frsb_kac-rice.tex | 22 |
1 files changed, 13 insertions, 9 deletions
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index 09c3042..1427085 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -52,11 +52,15 @@ replica-symmetry breaking scheme that is well-defined, and corresponds directly to the topological characteristics of those minima. -A more general question, of interest in optimization problems, is how to define a `threshold level'. This notion was introduced in Ref \cite{cugliandolo1993analytical} in the context of the $p$-spin model, as the energy at which the constant energy patches of phase-space percolate - hence -explaining why dynamics should relax to that level. +Perhaps the most interesting application of this computation is in the context of +optimization problems, see for example \cite{gamarnik2021overlap,alaoui2022sampling,huang2021tight}. A question +that appears there is how to define a `threshold level'. This notion was introduced \cite{cugliandolo1993analytical} in the context of the $p$-spin model, as the energy at which the patches of the same energy in phase-space percolate - hence +explaining why dynamics never go below that level. The notion of a `threshold' for more complex landscapes has later been -attempted several times, never to our knowledge in a clear and unambiguous -way. One of the purposes of this paper is to +invoked several times, never to our knowledge in a clear and unambiguous +way. One of the purposes of this paper is to give a sufficiently detailed +characterization of a general landscape so that a meaningful general notion +of threshold may be introduced - if this is at all possible. \section{The model} @@ -968,9 +972,9 @@ P(q)=\frac1{\mathcal N^2}\sum_{\mathbf s_1\in\mathcal S}\sum_{\mathbf s_2\in\mat {\em This is the probability that two stationary points randomly drawn from the ensemble of stationary points happen to be at overlap $q$} -It is -straightforward to show that moments of this distribution are related to -certain averages of the form. These are evaluated for a given energy, index, etc, but +%It is straightforward to show that moments of this distribution are related to +%certain averages of the form. +These are evaluated for a given energy, index, etc, but we shall omit these subindices for simplicity. \begin{equation} @@ -1027,8 +1031,8 @@ Consider two independent pure $p$ spin models $H_{p_1}({\mathbf s})$ and $H_{p_2 {\mathbf \sigma} \cdot {\mathbf s}$. The complexities are \begin{eqnarray} - e^{N\Sigma(e)}&=&\int de_1 de_2 \; e^{N[ \Sigma_1(e_1) + \Sigma_2(e_2) + O(\varepsilon) -\lambda N [(e_1+e_2)-e]}\nonumber \\ - e^{-G(\hat \beta)}&=&\int de de_1 de_2 \; e^{N[-\hat \beta e+ \Sigma_1(e_1) + \Sigma_2(e_2) + O(\varepsilon) -\lambda N [(e_1+e_2)-e]} + e^{N\Sigma(e)}&=&\int de_1 de_2 d\lambda \; e^{N[ \Sigma_1(e_1) + \Sigma_2(e_2) + O(\varepsilon) -\lambda N [(e_1+e_2)-e]}\nonumber \\ + e^{-G(\hat \beta)}&=&\int de de_1 de_2 d\lambda\; e^{N[-\hat \beta e+ \Sigma_1(e_1) + \Sigma_2(e_2) + O(\varepsilon) -\lambda N [(e_1+e_2)-e]} \end{eqnarray} The maximum is given by $\Sigma_1'=\Sigma_2'=\hat \beta$, provided it occurs in the phase in which both $\Sigma_1$ and $\Sigma_2$ are non-zero. The two systems are `thermalized', |